Let us now return to the general downlink AWGN channel without the symmetry assumption and take |h1| < |h2|. Now user 2 has a better channel than user 1 and hence can decode any data that user 1 can successfully decode. Thus, we can use the superposition coding scheme: First the transmit signal is the (linear) superposition of the signals of the two users. Then, user 1 treats the signal of user 2 as noise and decodes its data fromy1. Finally, user 2 which has the better channel performs SIC: it decodes the data of user 1 (and hence the transmit signal corresponding to the user 1’s data) and then proceeds to subtract the transmit signal of user 1 from y2 and decode its’ data. As before, with each possible power split of P =P1+P2, the following rate pair can be achieved:
R1 = log à
1 + P1|h1|2 P2|h1|2+N0
ả
bits/s/Hz R2 = log
à
1 + P2|h2|2 N0
ả
bits/s/Hz. (6.22)
On the other hand, orthogonal schemes achieve, for each power split P =P1+P2 and degree-of-freedom splitα ∈[0,1], as in the uplink (c.f. (6.8) and (6.9)),
R1 = αlog à
1 + P1|h1|2 αN0
ả
bits/s/Hz, R2 = (1−α) log
à
1 + P2|h2|2 (1−α)N0
ả
bits/s/Hz. (6.23)
y
y
^ x
2
x
x x
x
^ x
1 x
Figure 6.8: Superposition decoding example. The transmitted constellation point of user 1 is decoded first, followed by decoding of the constellation point of user 2.
Here, α represents the fraction of the bandwidth devoted to user 1. Figure 6.9 plots the boundaries of the rate regions achievable with superposition coding and optimal orthogonal schemes for the asymmetric downlink AWGN channel (with SNR1 = 0 dB and SNR2 = 20 dB). We observe that the performance of the superposition coding scheme is better than that of the orthogonal scheme.
One can show that the superposition decoding scheme is strictly better than the orthogonalization schemes (except for the two corner points where only one user is being communicated to), i.e., for any rate pair achieved by orthogonalization schemes there is a power split for which the successive decoding scheme achieves rate pairs that are strictly larger (see Exercise 6.26). This gap in performance is more pronounced when the asymmetry between the two users deepens. In particular, superposition coding can provide a very reasonable rate to the strong user, while achieving close to the single-user bound for the weak user. In Figure 6.9, for example, while maintaining the rate of the weaker userR1 at 0.9 bits/s/Hz, superposition coding can provide a rate of aroundR2 = 3 bits/s/Hz to the strong user while an orthogonal scheme can provide a rate of only around 1 bits/s/Hz. Intuitively, the strong user, being at high SNR, is degree-of-freedom limited and superposition-coding allows it to use the full degrees of freedom of the channel while being allocated only a small amount of transmit power, thus causing small amount of interference to the weak user. In contrast, an orthogonal scheme has to allocate a significant fraction of the degrees of freedom to the weak user
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0
1 2 3 4 5 6 7
Rate Of User 1
Rate of User 2
Figure 6.9: The boundary of rate pairs (in bits/s/Hz) achievable by superposition coding (solid line) and orthogonal schemes (dashed line) for the two user asymmetric downlink AWGN channel with the user SNRs equal to 0 and 20 dB (i.e.,P|h1|2/N0 = 1 and P|h2|2/N0 = 100). In the orthogonal schemes, both the power split P = P1+P2 and split in degrees of freedom α are jointly optimized to compute the boundary.
to achieve near single-user performance, and this causes a large degradation in the performance of the strong user.
So far we have considered a specific signaling scheme: linear superposition of the signals of the two users to form the transmit signal (c.f. (6.19)). With this specific encoding method, the SIC decoding procedure is optimal. However, one can show that this scheme in fact achieves the capacity and the boundary of the capacity region of the downlink AWGN channel is given by (6.22) (see Exercise 6.27).
While we have restricted ourselves to two users in the presentation, these results have natural extensions to the general K user downlink channel. In the symmetric case |h1|=|h2|=|h|, the capacity region is given by the single constraint
XK
k=1
Rk<log à
1 + P|h|2 N0
ả
. (6.24)
In general with the ordering |h1| ≤ |h2| ≤ ã ã ã ≤ |hK|, the boundary of the capacity region of the downlink AWGN channel is given by the parameterized rate tuple
Rk= log
1 + Pk|hk|2 N0+³PK
j=k+1Pj
´
|hk|2
, k = 1. . . K, (6.25)
where P = PK
k=1Pk is the power splits among the users. Each rate tuple on the boundary, as in (6.25), is achieved by superposition coding.
Since we have a full characterization of the tradeoff between the rates at which users can be jointly reliably communicated to, we can easily derive specific scalar per- formance measures. In particular, we focused on sum capacity in the uplink analysis;
to achieve the sum capacity we required all the users to transmit simultaneously (we employed the SIC receiver to decode the data). In contrast, we see from (6.25) that the sum capacity is achieved by transmitting to asingleuser, the user with the highest SNR.
Summary 6.1 Uplink and Downlink AWGN Capacity
Uplink:
y[m] = XK
k=1
xk[m] +w[m] (6.26)
with user k having power constraint Pk. Achievable rates satisfy:
X
k∈S
Rk ≤log à
1 + P
k∈SPk N0
ả
for all S ⊂ {1, . . . , K} (6.27)
The K! corner points are achieved by SIC, one corner point for each cancellation order. They all achieve the same optimal sum rate.
A natural ordering would be to decode starting from the strongest user first and move towards the weakest user.
Downlink:
yk[m] =hkx[m] +wk[m], k= 1, . . . K (6.28) with |h1| ≤ |h2| ≤. . .≤ |hK|.
The boundary of the capacity region is given by the rate tuples:
Rk= log
1 + Pk|hk|2 N0+³PK
j=k+1Pj
´
|hk|2
, k = 1. . . K, (6.29) for all possible splits P =P
kPk of the total power at the base station.
The optimal points are achieved by superposition coding at the transmitter and SIC at each of the receivers.
The cancellation order at every receiver is always to decode the weaker users before decoding its own data.
Discussion 6.9: SIC: Implementation Issues
We have seen that successive interference cancellation plays an important role in achieving the capacities of both the uplink and the downlink channel. In contrast to the receivers for the multiple access systems in Chapter 4, SIC is a multiuser receiver. Here we discuss several potential practical issues in using SIC in a wireless system.
• Complexity scaling with the number of users: In the uplink, the base station has to decode the signals of every user in the cell, whether it uses the conventional single-user receiver or the SIC. In the downlink, on the other hand, the use of SIC
at the mobile means that it now has to decode information intended for some of the other users, something it would not be doing in a conventional system. Then the complexity at each mobile scales with the number of users in the cell; this is not very acceptable. However, we have seen that superposition coding in conjunction with SIC has the largest performance gain when the users have very disparate channels from the base station. Due to the spatial geometry, typically there are only a few users close to the base station while most of the users are near the edge of the cell. This suggests a practical way of limiting complexity: break the users in the cell into groups, with each group containing a small number of users with disparate channels. Within each group, superposition coding/SIC is performed, and across the groups, transmissions are kept orthogonal. This should capture a significant part of the performance gain.
• Error propagation: Capacity analysis assumes error-free decoding but of course with actual codes, errors are made. Once an error occurs for a user, all the users later in the SIC decoding order will very likely be decoded incorrectly. Exercise 6.12 shows that if p(i)e is the probability of decoding the ith user incorrectly, assuming that all the previous users are decoded correctly, then the actual error probability for the kth user under SIC is at most
Xk
i=1
p(i)e . (6.30)
So, if all the users are coded with the same target error probability assuming no propagation, the effect of error propagation degrades the error probability by a factor of at most the number of users K. If K is reasonably small, this effect can easily be compensated by using a slightly stronger code (by, say, increasing the block length by a small amount).
• Imperfect channel estimates: To remove the effect of a user from the ag- gregate received signal, its contribution must be reconstructed from the decoded information. In a wireless multipath channel, this contribution depends also on the impulse response of the channel. Imperfect estimate of the channel will lead to residual cancellation errors. One concern is that, if the received powers of the users are very disparate (as in the example in Figure (6.3) where they differ by 20 dB), then the residual error from cancelling the stronger user can still swamp the weaker user’s signal. On the other hand, it is also easier to get an accurate channel estimate when the user is strong. It turns out that these two effects compensate each other and the effect of residual errors does not grow with the power disparity (see Exercise 6.13).
• Analog-to-digital quantization error: When the received powers of the users are very disparate, the analog-to-digital (A/D) converter needs to have a very large dynamic range, and at the same time, enough resolution to quantize accurately the contribution from the weak signal. For example, if the power disparity is 20dB, even 1-bit accuracy for the weak signal would require a 8-bit A/D converter. This may well pose an implementation constraint on how much gain SIC can offer.